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Statistical Estimations of the Distribution of Fire Growth Factor – Study on Risk-Based Evacuation Safety Design Method
YOSHIKAZU DEGUCHI1, HIROAKI NOTAKE2, JUN’ICHI YAMAGUCHI
3, and TAKEYOSHI TANAKA4 1Research & Development Institute Takenaka Corporation 1-5-1 Ohtsuka, Inzai-city, Chiba, 270-1395, Japan 2Institute of Technology Shimizu Corporation 3-4-17 Etchujima, Koto-ku, Tokyo, 135-8530, Japan 3Technical Research Institute Obayashi Corporation 4-640, Shimokiyoto, Kiyose-city, Tokyo, 204-8558, Japan 4Disaster Prevention Research Institute Kyoto University Gokasyo, Uji, Kyoto, 611-0011, Japan
ABSTRACT
In performance-based evacuation safety design methods at present, consideration of fire risk has been
insufficient. In order to develop the framework of performance-based evacuation safety verification, the
authors have proposed a methodology for selecting both design fire scenarios and design fires based on the
fire risk in fire, which we call the Risk-Based Evacuation Safety Design Method.
In the Risk-Based Evacuation Safety Design Method the fire growth factor α and its distribution f(α) is one
of the key points. We analyze the data of fire growth time (t) and burned area (Af) in the national fire
statistics in Japan to obtain distributions of fire growth factor in real fire situations in terms of burned area.
Then we estimate a heat release rate per unit floor area using the existing fire load density survey and burn
experiments. The distributions of fire growth factor f(α) are obtained by multiplying the burned area growth
factor (Af/t2) and the heat release rate per unit floor area (q'') .
It is found that the distributions of the fire growth factor f(α) follows a log-normal distribution and that the
fire growth factors from the existing experiments are within a reasonable range of the distributions.
KEYWORDS: risk-based design, fire growth factor, fire growth time, burned area, burned area growth
factor, heat release rate per unit floor area.
NOMENCLATURE LISTING
Af burned area (m2) fire growth factor (kW/s
2)
C causality (person) c limit fire growth factor (kW/s2)
C0 initial number of occupants (person) μ mean
D diameter of fire source (m) σ standard deviation
H flame height (m) λ mean of L-N distribution
P probability of loss by fire ζ standard deviation of L-N distribution
Q heat release rate (kW) subscripts R evacuation safety failure risk A acceptable
W fire load density (kg/m2) D design
q'' heat release rate per unit floor area (kW/m2) f start spraying
q''s heat release rate per unit surface area (kW/m2) i scenario number
t time (s) max maximum
Greek s start of fire
eff Effective surface area (m2/kg)
FIRE SAFETY SCIENCE-PROCEEDINGS OF THE TENTH INTERNATIONAL SYMPOSIUM, pp. 1087-1100 COPYRIGHT © 2011 INTERNATIONAL ASSOCIATION FOR FIRE SAFETY SCIENCE / DOI: 10.3801/IAFSS.FSS.10-1087
1087
INTRODUCTION
In Japan, performance-based codes have been introduced by the amendment of the Building Standard Law
in 2000 to provide more flexibility and clarity in the building regulatory system. However, the
consideration of fire risk has scarcely integrated. In the performance-based verification method, we use
uniform fire growth factors and criteria regardless of the size of spaces and occupants. This has caused
unbalanced practices in fire safety verification, such as that it is more difficult to verify the safety of small
rooms with insignificant numbers of occupants than the safety of whole building with a large number of
occupants. On the other hand, risk consideration is thought to be integrated into the existing prescribed
code even though it is implicit and empirical. The fire safety standards for large buildings and buildings
accommodating unspecified members of the public are implicitly more severe than the standards for other
buildings. In order to develop a sound framework for performance-based evacuation safety verification, the
authors proposed a methodology for selecting both design fire scenarios and design fires based on fire risk,
which we call the Risk-Based Evacuation Safety Design Method [1–5] but the frequency distribution of the
fire growth factor used was only an assumption.
In this paper, we analyze fire reports collected by the National Fire Defense Agency for fire statistics [6] to
obtain the frequency distribution of the fire growth factor α, The frequency distributions were obtained for
building uses of office, residence, restaurant and merchandise store.
CONCEPT OF RISK-BASED EVACUATION SAFETY DESIGN METHOD
Procedure of Fire Safety Verification
The Risk-Based Evacuation Safety Design Method is assumed to be conducted as the usual performance-
based evacuation safety design method procedure as illustrated in Fig. 1, in which evacuation safety is
verified by deterministic manner, where the fire conditions predicted under the selected design fires and
fire scenarios are compared with prescribed safety criteria.
The other important features of the current performance-based fire safety design methods to be taken into
account in the Risk-Based Evacuation Safety Design Method are:
Assumed fires are always hazardous fires, disregarding trivial fires such as smoldering fires,
Number of evacuees in a space is calculated assuming the space is fully loaded by occupants,
Safety criteria is set at strict level, such as „not exposed‟ or „only slightly exposed‟ to smoke,
Safety verification, including calculation methods used, is to be made conservative enough.
Evacuation Design
Fire Control Design
Smoke Control Design
?
yes
no
Determination of Design Fire Scenario i
Calculation of Design
Fire Growth Factor αD(i)
Prediction of Fire and Evacuation
Verification Start
Calculation Methods
RA(i): Acceptable Evacuation
Failure Risk
λ, ζ : Mean and Standard Deviation
Pi: Probability of Loss by Fire
C0: Initial Number of Occupants
0)( iC D
Safety Criteria
Verification End
Calculation of Design Acceptable
Evacuation Failure Risk RDA
Determination of Design Acceptable
Evacuation Failure Risk RDA(i) under the
scenario i . D
A
D
iA RR )(
Fig. 1. Procedure of Risk-Based Evacuation Safety Design Method.
1088
Relationship between Design Fire and Fire Growth Factor
Generally, the fire growth factor α depends on a variety of factors, such as, the type and intensity of
ignition source, size and material properties of ignited item and so on. Therefore, the growth factor of a fire
which starts in a building space, where various combustibles exist together cannot be constant but can vary
at random.
This risk-based design method considers that the fire growth factor α follows a probability density function
f(α) which is set according to building use. Besides, because of targeting on floor evacuation safety, at the
initial stage of fire, the heat release rate (HRR) of the fire source is assumed to increase in proportion to
time-squared (t2 fire) and becomes constant after it reaches the maximum HRR, as in many other
evacuation safety verification methods at present [7].
The maximum HRR is controlled by ventilation conditions, sprinkler installation etc. and is assumed to be
constant although, in reality, it may vary depending on the various conditions involved.
Relationship between Acceptable Evacuation Failure Risk and Fire Growth Factor
Figure 2 shows a relationship between the acceptable evacuation failure risk and fire growth factor in the
context of the Risk-Based Evacuation Safety Design Method.
Generally, the number of causalities to occur depends on the magnitude of the fire growth factor .
Figure 2 (top left) shows a concept of the relationship between fire growth factor and causalities caused
C(). Causalities will be zero when the fire growth factor is very small but will begin to increase when
the fire growth factor exceeds a certain limit fire growth factor c and it becomes constant at C0 when
further increases. It is too difficult, or impossible in nature, to know how C() increases in the range of >
c, because it depends on many conditions related with evacuation safety. However, since causalities C()
cannot exceed the initial occupant load but a fire with the limit fire growth factor c may become large.
That is, C() = C0( > c) is a conservative evaluation. Therefore, the design evacuation risk RD(i) is
described as Eq. 1 under the fire scenario i, letting Pi be probability of loss of fire.
c
dfCPiR i
D
0
(1)
A design goal of the Risk-Based Evacuation Safety Design Method is to control the evacuation risk under
an acceptable risk level, i.e., RD < RA
D. Therefore, for an evacuation safety plan to be acceptable, reading
the critical fire growth factor c in Eq. 1as the design fire growth factor D, it is necessary to design the
spaces and safety systems to allow all the occupants to evacuate safely under the fire with (0 < < D).
iRdfCP D
AiD
0
(2)
In other words, the evacuation safety can be accomplished by the design that allows no casualties to occur
(C() = 0) under the design fire Q = D t2, which can be obtained by solving.
0CP
iRdf
i
D
A
D
(3)
1089
C
Pro
babi
lity
dens
ity f(
)C
asua
lties
C
Fire growth factor
unclear
C0
Ris
k dis
tribu
tion
Occurrence probability
CasualtiesC=C0 > C)
Evacuation failure risk
D
Fire growth factor
Fire growth factor
Design acceptable evacuation failure risk RD
A
C
Design evacuation failure risk RD
Fig. 2. Relationship between acceptable evacuation failure risk and fire growth factor.
ESTIMATION METHOD OF THE DISTRIBUTION OF FIRE GROWTH FACTOR
Fire Growth Factor in Verification Method of Evacuation in Japan
Since oxygen consumption calorimetric technology was invented, burning characteristics of various items
have been measured and their fire growth factors have been obtained. However, it is still not known what
items and how frequently they are ignited in real fire situations. In performance-based fire safety design
practices, the fire growth factor of design fires is selected based on expert discretion, often from NFPA‟s
fast, medium, slow fires.
In Japanese procedure for evacuation safety design, Calculation method etc. for the verification method for
floor Evacuation safety [7], the fire growth factor is correlated with the calorific fire load density (ql) as
follows,
>
170106.2
1700125.03/56
ll
l
q (4)
Equation 4 is based on a model which takes into account the ideal conditions that homogeneous
combustibles are uniformly distributed in a space. Although this formula is actually used in the current
verification of evacuation safety of buildings, the basic assumption of Eq. 4 is not necessarily reasonable
enough to be defensible. However, more preferably, the fire growth factor is estimated with fire statistics
which represent actual fire conditions.
Calculation Method for the Fire Growth Factor Based on Fire Statistics
Fire statistics reflects real fire situations better than burn experiments but, needless to say, limitations exist
to in using fire statistics to obtain the fire growth factor . Firstly, HRR is not measured in real fire, so we
need to be satisfied with obtaining burned area instead. Therefore HRR per unit burned area have to be
estimated by some other means. In this paper, considering the formula as below, we obtain the fire growth
factor in terms of burned area, Af, from fire reports, and HRR per unit burned area q'' from fire load density
survey and burn experiments separately, and obtain the fire growth factor by multiplying these two.
1090
The same method for obtaining fire growth factor using burned area from fire statistics was applied by
Charters et al. to limited occupancy [8,9].
22t
Aq
tt
Q f
sf
f
(5)
where, Qf is the HRR at start of water spraying time(kW), q'' is the HRR per m2 (kW/m
2), Af is the burned
area (m2), tf is the start of water spraying time (s), ts is the ignition time (s) and t = tf - ts.
ESTIMATION OF THE DISTRIBUTION OF BURNED AREA GROWTH FACTOR BASED ON
FIRE STATISTICS
Fire Statistics Data
The data used in this paper to analyze the burn area growth factor are the fire reports submitted to the
National Fire Defense Agency for national fire statistics. Of these, we extracted the fire reports of the
buildings with uses office, residence, restaurant and merchandise store considering the amount of data
available for the analyses. The data were limited focusing on the initial stage of fire considering that the
purpose of the survey is to analyze the initial fire growth factor. Also, a small number of extremely fast
developing fires, due to explosion and the like, are excluded.
The attributes of the data extracted from the fire reports are as follows:
year; 1995–2008,
number of damaged buildings; 1,
burned area; more than 1 m2,
first aid fire fighting ; none,
time from start spraying to suppression ; less than 60 min,
time from break out to start spraying ; less than 20 min,
burned area is under 90 % of floor area of fire origin,
building use on fire; office, residence, restaurant, merchandise store,
exclude fire growth factor is over 1.
Treatment of Fire Statistics Data
Figure 3 illustrates the method of handling the fire report data to calculate the fire growth factor . The
dashed line shows the image of growth of burned area Af in actual fire and the solid line shows the image of
approximated fire growth curve. The symbol, ★ stands for the data that is available from the fire report.
estimatedignitiontime
time(min.)
Bu
rne
d A
rea
(m2)
start of waterspraying time
undercontrolledtime
Af
Image of real fire behavior
t
A
D
C
F
completelyextinguishedtime
first calltime
ignitiontime
B
E
assumed fire behavior
Fig. 3. Image of treatment of fire statistics.
1091
The fire ignition time (C) was estimated by the fire department. Although a certain degree of error may be
involved between the true values (A), we trust the accuracy of the professional estimate by the fire
department. While the start of water spraying time (B) must be almost accurate, we cannot obtain the
burned area at that time (D). Alternately, we replace the time (B) with that of the final burned area (F).
Assuming that the fire spread was negligible after the start of water spraying because we focus the early
fire stage, so that the burned area at the start water spraying (E) is not greatly different from the burned area
at the time of extinguishment (F).
Analysis of Extracted Data
Fire prevention measures are classified into four categories, „single use‟, „multiple uses‟, „small scale‟ and
„others‟. Figure 4 shows the fire prevention measures categories by each building use. Figure 5 shows the
categories of causes of fire and Fig 6 shows the categories of fire source. The important features are:
In the case of office, „Cigarettes‟ and „Arson‟ have a high proportion in causes of fire and „Fibers
Combustibles‟ and „Wastes‟ have a high proportion in fire source excluding „unknown‟.
In the case of residence, „Cigarettes‟ has a high proportion of causes of fire and „Fibers
Combustibles‟ accounts for 40 %.
In the case of restaurant, causes of fire are variety, although „Cook appliances‟ have a high
proportion compared with other building uses and „Inflammable Liquid‟ has high proportion in
fire source compared with other building uses.
In the case of merchandise store, causes of fire are similar to that of restaurant and „Solid
Combustibles‟ has a high proportion in fire source compared with other building uses.
n=601
Small-scale21%
Others18% (15)
(Work place)38%
(16)a,b(Multiple uses)
24% n=11,598
Small-scale62%
Other1%
(5)b(Apartments)
28%
(16)a,b(Multiple uses)
19%
n=564
Small-scale21%
Others6%
(3)b(Eating and
drinkong houses)29%
(16)a,b(Multiple ises)
43%
n=313
Small-scale17%
Other4%
(4)(Department
stores)36%
(16)a,b(Multiple uses)
43%
(a) (b) (c) (d)
Fig. 4. Categories of building use by fire prevention measures: (a) office; (b) residence; (c) restaurant; (d)
merchandise store.
0% 20% 40% 60% 80% 100%
Office
Residence
Restaurant
Store
Cigarettes Cook appliances Heating equipmentsElectrical devices Wiring Play with fireOpen flame Incinerator ArsonOthers Unknown
Fig. 5. Categories of causes of fire.
1092
0% 20% 40% 60% 80% 100%
Office
Residence
Restaurant
Store
Walls and Structual frames FlooringsFurnitures & Interior goods Inflammable LiquidSorid Combustibles Fibers CombustiblesWastes OthersUnknown
Fig. 6. Categories of fire source.
Distribution of Start of Water Spraying Time
Figure 7 shows the relative frequency of the time from ignition to start of water spraying t. As Fig. 7
indicates, the mean (12.9–13.2) and standard deviation (3.7–4.0) of start of water spraying time t are
similar with each building use.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0 2 4 6 8
10
12
14
16
18
20
Re
lati
ve F
req
ue
ncy
t(min.)
μ=13.0σ=3.8
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0 2 4 6 8
10
12
14
16
18
20
Re
lati
ve F
req
ue
ncy
t(min.)
μ=13.2σ=3.7
(a) (b)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0 2 4 6 8
10
12
14
16
18
20
Re
lati
ve F
req
ue
ncy
t(min.)
μ=13.0σ=4.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0 2 4 6 8
10
12
14
16
18
20
Re
lati
ve F
req
ue
ncy
t(min.)
μ=12.9σ=3.8
(c) (d)
Fig. 7. Relative frequency of start of water spraying time t: (a) office; (b) residence; (c) restaurant;
(d) merchandise store.
1093
Distribution of Burned Area
Figure 8 shows the relative frequency of burned area Af. As Fig. 8 indicates, the mean burned area of
residence and merchandise store are larger than the others. However, the frequency distributions are similar
for all the types of use.
0.00
0.05
0.10
0.15
0.20
0.25
0
10
20
30
40
50
60
70
80
90
10
0
11
0
12
0
13
0
14
0
15
0
Re
lati
ve F
req
ue
ncy
Af(m2)
μ=35.6σ=34.2
0.00
0.05
0.10
0.15
0.20
0.25
0
10
20
30
40
50
60
70
80
90
10
0
11
0
12
0
13
0
14
0
15
0
Re
lati
ve F
req
ue
ncy
Af(m2)
μ=42.5σ=35.3
(a) (b)
0.00
0.05
0.10
0.15
0.20
0.25
0
10
20
30
40
50
60
70
80
90
10
0
11
0
12
0
13
0
14
0
15
0
Re
lati
ve F
req
ue
ncy
Af(m2)
μ=39.7σ=36.0
0.00
0.05
0.10
0.15
0.20
0.25
0
10
20
30
40
50
60
70
80
90
10
0
11
0
12
0
13
0
14
0
15
0
Re
lati
ve F
req
ue
ncy
Af(m2)
μ=45.8σ=44.5
(c) (d)
Fig. 8. Relative frequency of burned area Af: (a) office; (b) residence; (c) restaurant; (d) merchandise store.
Distribution of Burned Area Growth Factor
The burned area growth factors Af/t2 were obtained and statistically processed. Table 1 shows the statistics:
mean , standard deviation , log-normal distribution‟s mean and standard deviation , defined by Eq. 6
and 7. Figure 9 illustrates the relative frequency from the fire statistics and the log-normal distribution
function shown in Table 1.
22
logexp
2
1
g (6)
2
2
1ln ,
2
22 1ln
(7)
From Table 1 and Fig. 9, the building uses do not make much difference in the mean and standard
deviation . However, restaurant is the smallest and merchandise store is the largest mean and standard
deviation, although causes of fire are similar in restaurant and merchandise store, as we noted in the
analysis of Fig. 5. This tendency may be attributed to the fact that only somewhat limited types of
combustibles (not causes of fire and fire source) tend to exist in a restaurant relative to a merchandise store
where more variety of combustibles exists.
1094
In all building uses, the data of fire statistics is larger than the assumed log-normal distributions in the
range of 5.0 × 10-5
. In other words, the statistics include the fires which spread more slowly than the
assumed log-normal distribution. This means it is more conservative to use the assumed log-normal
distribution in the evacuation verification. Therefore, the frequency distribution of the burned area growth
factor Af/t2 can be modeled as a log-normal distribution.
Table 1. Statistics of burned area growth factor.
Building use Office Residence Restaurant Store
Number 601 11,598 564 313
Burned area growth factor Af/t2 (×10
-5 m/s
2)
mean 7.6 8.1 7.4 9.7
standard deviation 11.0 9.3 7.8 12.2
minimum 0.077 0.069 0.077 0.069
75 percentile 3.9 5.4 5.0 5.3
90 percentile 9.4 10.4 10.2 12.3
95 percentile 18.3 17.4 17.0 22.6
99 percentile 57.2 45.4 37.3 59.5
maximum 224.0 131.1 45.3 70.9
log-normal
distribution
-9.15 -10.85 -9.89 -9.72
1.19 0.92 0.87 0.97
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
Cu
mu
lati
ve P
rob
abili
ty(%
)
Re
lati
ve F
req
ue
ncy
Af/t2 (×10-5 m2/s2)
μ=7.6×10-5
σ=11.0×10-5
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.0
2.5
5.0
7.5
10
.0
12
.5
15
.0
17
.5
20
.0
22
.5
25
.0
27
.5
30
.0
Cu
mu
lati
ve P
rob
abili
ty(%
)
Re
lati
ve F
req
ue
ncy
Af/t2 (×10-5 m2/s2)
μ=8.1×10-5
σ=9.3×10-5
(a) (b)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.0
2.5
5.0
7.5
10
.0
12.5
15.0
17.5
20.0
22.5
25.0
27.5
30.0
Cu
mu
lati
ve P
rob
abili
ty(%
)
Re
lati
ve F
req
ue
ncy
Af/t2 (×10-5 m2/s2)
μ=7.4×10-5
σ=7.8×10-5
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.0
2.5
5.0
7.5
10
.0
12
.5
15
.0
17
.5
20
.0
22
.5
25
.0
27
.5
30
.0
Cu
mu
lati
ve P
rob
abili
ty(%
)
Re
lati
ve F
req
ue
ncy
Af/t2 (×10-5 m2/s2)
μ=9.7×10-5
σ=12.2×10-5
(c) (d)
Fig. 9. Frequency distribution of burn area growth factor Af/t2: (a) office; (b) residence; (c) restaurant;
(d) merchandise store.
1095
ESTIMATION OF THE HEAT RELEASE RATE PER UNIT FLOOR AREA
Estimation Method
As described above, it is necessary to know the HRR per unit floor area q'' to obtain the fire growth factor
(kW/s2). The q'' is influenced by the combustible conditions. In this paper, the following two methods are
tentatively proposed serves as examples to estimate q'':
EXAMPLE 1: Estimation from combustible survey
EXAMPLE 2: Estimation from burn experiments
However, there may be more accurate estimation methods depending on the availability of relevant data.
EXAMPLE 1: Estimation from Combustible Survey
Kurioka et al. [10] have investigated combustible distribution, fire load and combustible surface area in
standard office buildings. Equation 8 shows the relationship between the effective surface area eff and the
fire load density W. In this paper, it is reported that the effective surface area eff, is proportional to
the -2/3rd
power of the fire load density W, which converts synthetic high flammability polymers into a
wood equivalent in terms of calorific value. Table 2 shows the fire load density W in this survey.
32
70.0
Weff (8)
where, eff is the effective surface area(m2[fuel]/kg), W is the fire load density which is the converted
synthetic high flammability polymer into wood (kg/m2[floor]).
Table 2. Results on fire load density survey.
Building use Fire load density W (kg/m
2)
range mean
Clerical desk unit 22.4–52.0 38.9
Engineering desk unit 43.3–82.8 65.6
Here, assuming the heat release rate of fire source Q can be related to the combustible surface area, q'' can
be given as a function of the fire load density W as Eq. 9.
31
70.0 WqWqq seffs (9)
where, qs'' is the heat release rate per unit fuel surface area (kW/m2[fuel]).
According to the existing test data, the HRR per unit fuel surface area of wood based fuel q'' ranges from
48–112 (kW/m2). Here we adopt 112 kW/m
2 as a conservative estimation. From Eq. 9 and Table 2, the
required q'' can be obtained from Table 3. If the fire load density W is already known, it will be convenient
to use this method.
Table 3. Estimated HRR per m2 based on fire load density survey.
Building use
Heat release rate per m2
q'' (kW/m2)
range mean
Clerical desk unit 218.7–288.0 262.4
Engineering desk unit 271.9–336.7 311.8
1096
EXAMPLE 2: Estimation from Burn Experiments
In order to obtain q'' from burn experiments, the values of both heat release rate Q and burned area Af have
to be measured. However, information for burned area Af is rarely available in almost all of experiments
performed in the past. In this paper, a value of q'' is obtained from the results of a previous burn
experiments known as the Kuramae fire experiment [11], which includes information on the burned area Af.
The heat release rate Q is calculated from the flame height measured in the experiment.
The Kuramae fire experiment was conducted at the old Sumo Arena by the Tokyo Fire Department to
investigate the smoke layer filling behavior and the performance of fire detectors for use in firefighting.
Assuming a merchandise store in a large compartment with a high ceiling, 600 kg of hanged cloths was
arranged in a 20 m2 area, i.e. 30 kg/m
2, as fuel, and ignited at the center by using 100 cc of methanol.
Calculation of HRR per Unit Floor Area q”
Figure 10 shows the burning properties: weight loss, burned area and the flame height, measured in the
Kuramae experiment. From Fig. 10, 7 min after ignition, the fire has reached a peak HRR, at which the
burned area Af was 20 m2 and the flame height H was 12 m. We estimate the heat release rate Q using
Eq. 10 for flame height:
DD
QH
32
2511165.3 (10)
where, H is flame height (m) and D is diameter of fire source (m).
Substituting H = 12 m, D = √20 m into Eq. 10, the peak heat release rate Qmax was found to be 31.7 MW,
and the HRR per unit floor area q” is calculated to be 1,585 kW/m2 as shown below.
585,120
700,31
fA
Qq (11)
0
5
10
15
20
0
200
400
600
800
0 5 10 15 20
Bu
rne
d A
rea(
m2 )
, Fla
me
He
igh
t(m
)
Wei
ght
(kg)
time(min.)
WeightBurned AreaFlame Height
Max 20 m2
Start spraying
Fig. 10. Measured burn properties in Kuramae fire experiment [11].
1097
CALCULATION OF DISTRIBUTION OF FIRE GROWTH FACTOR
The fire growth factor distribution f() is given by multiplying q'' and the distribution of burned area
growth factor g() from Eq. 12.
gqf (12)
where, g() is the probability density of the burned area growth factor Af/t2 demonstrated in Fig. 9.
The HRR per unit floor areas q'', are shown in Table 4. In an engineering office case, the value is
311.8 kW/m2
. Although the relationship between the effective surface area eff and the fire load density W
as described in Eq. 8 could not be acquired for general building uses except office, we regard that Eq. 8
applys to other type of use and estimate q'' using the calorific load density prescribed in Calculation
method etc. for the verification method for floor Evacuation safety.
Table 4. Assumed fire load density and estimated HRR per unit floor area.
Building use Fire load
density W (kg/m2)
HRR per m2
q'' (kW/m2)
Office 65.6 311.8
Residence 45.0 278.9
Restaurant 30.0 243.6
Merchandise store 30.0 243.6
The results of the calculation using the q'' in Table 4 are shown in Fig. 11. In addition, the fire growth
factor for merchandise store is also calculated using experimentally obtained q'' in Fig. 12.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.0
0
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
0.1
0
0.1
1
0.1
2
Co
mu
lati
ve P
rob
abil
ity(
%)
Re
lati
ve F
req
uen
cy
Fire Growth Factor α (kW/s2)
μ=0.0236σ=0.034(q=311.8)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.0
0
0.0
1
0.0
2
0.0
3
0.0
4
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
0.1
0
0.1
1
0.1
2
Co
mu
lati
ve P
rob
abil
ity(
%)
Re
lati
ve F
req
uen
cy
Fire Growth Factor α (kW/s2)
μ=0.0524σ=0.060(q=278.9)
(a) (b)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
Co
mu
lati
ve P
rob
abil
ity(
%)
Re
lati
ve F
req
uen
cy
Fire Growth Factor α (kW/s2)
μ=0.0184σ=0.021(q=243.6)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
Co
mu
lati
ve P
rob
abil
ity(
%)
Re
lati
ve F
req
uen
cy
Fire Growth Factor α (kW/s2)
μ=0.0236σ=0.030(q=243.6)
(c) (d)
Fig. 11. Frequency distribution of fire growth factor : (a) office; (b) residence; (c) restaurant;
(d) merchandise store.
1098
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Co
mu
lati
ve P
rob
abil
ity(
%)
Re
lati
ve F
req
uen
cy
Fire Growth Factor α (kW/s2)
μ=0.153σ=0.193(q=1585)
Fig. 12. Frequency distribution of fire growth factor using experimentally obtained q''.
In conjunction with the introduction of the performance-based evaluation method of fire service equipment
into the Fire Service Law, a series of experiments for the investigation of the performance of sprinkler
system were conducted [12]. Three kinds of fire source (small; watch/jewel store, middle; toy store, large;
furniture store) were set up in the experiment. The fire growth factors estimated from the experiments are
shown in Table 5.
The mean value of the fire growth factor for office = 0.0236 kW/s2 in Fig. 11 is within the range of 0.02
to 0.05 kW/s2 in Table 5. However the value of store use, 0.0236 kW/s
2, is much less than the values for the
experiment in Table 5. On the other hand, as shown in Fig. 12, the value of store use established with the
experimentally obtained q'' is 0.153 kW/s2, that is between the value of watch/jewel store (0.0776 kW/s
2)
and toy store (0.2129 kW/s2), which may be reasonable if small items burn more frequently than large
items in real fires. However, it may be necessary to use the distribution in Fig. 12 for shops selling cloths
and big furniture stores for conservative treatment.
Needless to say, direct comparison between the statistically obtained and experimentally obtained fire
growth factors is difficult. In real fires, a variety items can burn in an unintended manner while selected
items are intentionally burned in experiments. Therefore the comparison here is just to confirm the
agreement at an order of magnitude for the fire growth factors.
Table 5. Experimental result of fire growth factor [12].
Building use Fire growth factor (kW/s2)
Office 0.02–0.05
Merchandise store
(small; watch/jewel) 0.0776
(middle; toy) 0.2129
(large; furniture) 0.4210
CONCLUSION
In order to develop the framework for the Risk-Based Evacuation Safety Design Method the authors have
proposed a methodology for selecting both design fire scenarios and design fires based on fire risk.
In the Risk-Based Evacuation Safety Design Method, the fire growth factors and their distribution f() is
among the key points. Firstly we obtained the fire growth factors in terms of burn area by analyzing the fire
reports submitted for national fire statistics. Secondly, we estimated heat release rate per unit floor area (q'')
and burned area growth factor (Af/t2).
It is found that the estimated frequency distribution of the fire growth factor approximately follows the
curve of a log-normal distribution. In future, it may be necessary to further improve the selection of heat
release rate per unit floor area q'' to estimate the distribution of fire growth factor more accurately.
1099
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